Displaying similar documents to “Approximation of C ∞ -functions without changing their zero-set”

The trivial locus of an analytic map germ

H. Hauser, G. Muller (1989)

Annales de l'institut Fourier

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We prove: For a local analytic family { X s } s ∈ S of analytic space germs there is a largest subspace T in S such that the family is trivial over T . Moreover the reduction of T equals the germ of those points s in S for which X s is isomorphic to the special fibre X 0 .

On real algebraic links in S 3

R. Benedetti, M. Shiota (1998)

Bollettino dell'Unione Matematica Italiana

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Viene presentata una costruzione che, dato un arbitrario nodo L ⊆ S 3 , produce allo stesso tempo: 1) un'applicazione polinomiale f : R 4 , 0 → R 2 , 0 con singolarità (debolmente) isolata in 0 e L come tipo di nodo della singolarità; 2) una risoluzione delle singolarità di f nel senso di Hironaka. Specializzando la costruzione ai nodi fibrati otteniamo una versione debole (a meno di scoppiementi e nella categoria analitica reale) di un reciproco per il teorema di fibrazione di Milnor.

On the Pythagoras numbers of real analytic set germs

José F. Fernando, Jesús M. Ruiz (2005)

Bulletin de la Société Mathématique de France

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We show that (i) the Pythagoras number of a real analytic set germ is the supremum of the Pythagoras numbers of the curve germs it contains, and (ii) every real analytic curve germ is contained in a real analytic surface germ with the same Pythagoras number (or Pythagoras number 2 if the curve is Pythagorean). This gives new examples and counterexamples concerning sums of squares and positive semidefinite analytic function germs.

Topological invariants of analytic sets associated with Noetherian families

Aleksandra Nowel (2005)

Annales de l’institut Fourier

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Let Ω ⊂ ℝ n be a compact semianalytic set and let ℱ be a collection of real analytic functions defined in some neighbourhood of Ω . Let Y ω be the germ at ω of the set ⋂ f ∈ ℱ f - 1 ( 0 ) . Then there exist analytic functions v 1 , v 2 , ... , v s defined in a neighbourhood of Ω such that 1 2 χ ( lk ( ω , Y ω ) ) = ∑ i = 1 s sgn v i ( ω ) , for all ω ∈ Ω .

Overstability and resonance

Augustin Fruchard, Reinhard Schäfke (2003)

Annales de l’institut Fourier

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We consider a singularity perturbed nonlinear differential equation ε u ' = f ( x ) u + + ε P ( x , u , ε ) which we suppose real analytic for x near some interval [ a , b ] and small | u | , | ε | . We furthermore suppose that 0 is a turning point, namely that x f ( x ) is positive if x ≠ 0 . We prove that the existence of nicely behaved (as ϵ → 0 ) local (at x = 0 ) or global, real analytic or C ∞ solutions is equivalent to the existence of a formal series solution ∑ u n ( x ) ε n with u n analytic at x = 0 . The main tool of a proof is a new “principle of analytic continuation” for...